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Cell complex

A separable space that is a union of non-intersecting cells. Here, by a -dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension . If for each -dimensional cell of one is given a continuous mapping from the -dimensional cube into such that: 1) the restriction of to the interior of is one-to-one and the image is the closure in of (here is a homeomorphism of onto ); and 2) the set , where is the boundary of , is contained in the union of the cells of , then is called a cell complex; the union is called the skeleton of dimension of the cell complex . An example of a cell complex is a simplicial polyhedron.

A subset of a cell complex is called a subcomplex if it is a union of cells of containing the closures of such cells. Thus, the -dimensional skeleton of is a subcomplex of . Any union and any intersection of subcomplexes of are subcomplexes of .

Any topological space can be regarded as a cell complex — as the union of its points, which are cells of dimension 0. This example shows that the notion of a cell complex is too broad; therefore narrower classes of cell complexes are important in applications, for example the class of cellular decompositions or CW-complexes (cf. CW-complex).